Problem 77
Find the solution to the initial value problem. $$ x y^{\prime}=y(x-2), y(1)=3 $$
Problem 78
Find the solution to the initial value problem. $$ y^{\prime}=3 y^{2}(x+\cos x), y(0)=-2 $$
Problem 79
Find the solution to the initial value problem. $$ (x-1) y^{\prime}=y-2, y(0)=0 $$
Problem 80
Find the solution to the initial value problem. $$ y^{\prime}=3 y-x+6 x^{2}, y(0)=-1 $$
Problem 80
Substitute \(y=B e^{3 t}\) into \(y^{\prime}-y=8 e^{3 t}\) to find a particular solution.
Problem 81
Substitute \(y=a \cos (2 t)+b \sin (2 t)\) into \(y^{\prime}+y=4 \sin (2 t)\) to find a particular solution.
Problem 81
Draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field. $$ y^{\prime}=2 y-y^{2} $$
Problem 82
Draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field. $$ y^{\prime}=\frac{1}{x}+\ln x-y, \text { for } x>0 $$
Problem 82
Substitute \(y=a+b t+c t^{2}\) into \(y^{\prime}+y=1+t^{2}\) to find a particular solution.
Problem 83
Substitute \(y=a e^{t} \cos t+b e^{t} \sin t\) into \(y^{\prime}=2 e^{t} \cos t\) to find a particular solution.
